Chapter 7, Survey Sampling
Solution 61
(a)
We need $\, \Exp(\hat \beta) = \beta \,$. Since $\, \hat \beta = w_1 \hat X_1 + w_2 \hat X_2 + w_3 \hat X_3 \,$, we have $\, \Exp(\hat \beta) = (w_1 + w_2 + w_3) \alpha + (w_1 + 2 w_2 + 3 w_3) \beta \,$(Using $\Exp(\bar X_i) = \mu(t_i) = \alpha + \beta t_i$) should be equal to $\, \beta \,$. Thus $\, w_1 + w_2 + w_3 = 0 \,$ and $\, w_1 + 2 w_2 + 3 w_3 = 1 \,$.
(b)
TODO
$$\tag*{$\blacksquare$} $$